Optimal delocalized orbitals

We now investigate orbitals that range over both centers with linear combinations that minimize the calculated energy. For this simple two-electron system these may all be viewed as extensions of the Coulson-Fisher approach we describe next. We use the basis of Table 2.2 and compare these results with the appropriate full MC VB calculations of Section 2.8. 

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The first calculation of the energy of H2 for optimal delocalized orbitals used... 

As is well-known, modem valence-bond (VB) theory in its spin-coupled (SC) form (for a recent review, see Ref. 7) provides an alternative description of benzene [8-10] which, in qualitative terms, is no less convincing and is arguably even more intuitive than the MO picture with delocalized orbitals. The six n electrons are accommodated within a single product of six nonorthogonal orbitals, the spins of which are coupled in all five possible ways that lead to an overall six-electron singlet. The simultaneous optimization of the orbitals and of the weights of the five six-electron singlet spin... 

Fig. 5.8 Top direct (through space) hopping between two magnetic centers by tat with strongly locaUzed atomic orbittils Bottom effective (through Ugand) hopping by t"fj with self-consistently optimized magnetic orbitals, which have delocalization tails on the (bridging) ligands...

Cations are by no means the only species where the effects of hyperconjugative delocalization reveal themselves in such a striking manner. Similar effects exist in neutral systems or in anions. For instance, the normal propyl anion should tend to be eclipsed (E) since in this manner the molecule would optimize the 4-electron interactions between the ethyl group t orbital and the p orbital which carries the electron pair. In the bisected conformation, where ttchs and ttchs have both been raised in energy, the four-electron, destabilizing (see Section 1.7, rule 2) p ->7r interaction is stronger than in the eclipsed conformation. At the same time the two-electron, stabilizing p ->ir interaction is weaker than in the eclipsed conformation. Both effects favor the eclipsed conformation. 

Fe3+X6...Fe2+X6, which is the reactant of the outer-sphere electron transfer reaction mentioned above when X = Y. Clearly the ground state involves a symmetric linear combination of a state with the electron on the right (as written) and one with the electron on the left. Thus we could create the localized states by using the SCRF method to calculate the symmetric and antisymmetric stationary states and taking plus and minus linear combinations. This is reasonable but does not take account of the fact that the orbitals for non-transferred electrons should be optimized for the case where the transferred electron is localized (in contrast to which, the SCRF orbitals are all optimized for the delocalized adiabatic structure). The role of solvent-induced charge localization has also been studied for ionic dissociation reactions [109],... 

The most recent theoretical study, by Alhrichs and co-workers, deals with the di(phosphino)carbene Id and (phosphino) (phosphonio)carbenes Ie,f.16 The optimized geometry of the di(phosphino)carbene Id is weakly bent (PCP angle 160.5°) and highly unsymmetrical Only one of the phosphorus centers (P1) is in a planar environment, and it is much more closely bonded to the carbenic center than the other one (P1C 1.533 and P2C 1.765 A). The atomic charges (P1 +1.0, C -0.8, P2 +0.6) indicate that the short P bond is a double bond reinforced by Coulombic attraction, while the nature of the molecular orbitals revealed a slight delocalization of the carbene lone pair into the low-lying a (P-N) orbitals of the two phosphino substituents. The distortion from the symmetrical structure can be viewed as a second-order Jahn-Teller effect. 

Here, a and b are purely localized AOs, while Coulson—Fischer orbitals energy minimization, are generally not very delocalized (e < 1), and as such they can be viewed as distorted orbitals that remain atomic-like in nature. However, minor as this may look, the slight delocalization renders the Coulson—Fischer wave function equivalent to the VB-full wave function (Eq. 3.4a) with the three classical structures. A straightforward expansion of the Coulson—Fischer wave function leads to the linear combination of the classical structures in Equation 3.6. 

All the VB methods that deal with semilocalized orbitals use a generalization of the Coulson—Fischer idea (12), whereby a bond is described as a singlet coupling between two electrons in nonorthogonal orbitals that possess small delocalization tails resulting from the variational orbital optimization. Albeit formally covalent, this description implicitly involves some optimal contributions of ionic terms, as a decomposition of the wave function in terms of pure AO determinants would show (see Eqs. 3.5 and 3.6). For a polyatomic... 

The generalized valence bond (GVB) method was the earliest important generalization of the Coulson—Fischer idea to polyatomic molecules (13,14). The method uses OEOs that are free to delocalize over the whole molecule during orbital optimization. Despite its general formulation, the GVB method is usually used in its restricted form, referred to as GVB SOPP, which introduces two simplifications. The first one is the perfect-pairing (PP) approximation, in which only one VB structure is generated in the calculation. The wave function may then be expressed in the simple form of Equation 9.1, as a product of so-called geminal two-electron functions ... 

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